Two new recent posts take up the question of "spherical cows," the old joke term for absurd-sounding approximations that physicists make to turn intractable problems into easy ones. First, The First Excited State explains when N=N+1:
Everybody who's taken any sort of math class knows that a statement like N+1 = N is simply ridiculous. Everyone, that is, except for the physicist. Let's say that N is a really huge number, like if someone dumped an entire truckload of M&M's in your driveway. If you turned your back on me to watch the truck drive away, and I threw another M&M in the pile while you weren't looking, would you really notice? What if I snuck one while you were looking to the sky to thank God for this miracle? No, you'd really have no idea. So in this case, for all practical purposes, N+1 = N-1 = N. We make this approximation all the time in my statistical mechanics class, where N represents some astronomically huge number, like the number of water molecules in your glass.
Then the author links to a post at Shores of the Dirac Sea in which David Berenstein offers his own take:
Let me give you an example of how spherical cows might be put to some use. Assume that a cow is falling from a cliff onto a body of water like a lake (we don't want the cow to get hurt after all, we are not cruel). A physics question you might ask is how long does the cow take to fall. This is controlled by gravity and the height of the cliff. One of the things we know is that the shape of the cow does not really change the result much. Sure, there is some air resistance, etc, etc, but these effects are small. Saying an effect is small is stating that you only need an approximate answer and not an exact one. For these situations, given that you don't have the information of the exact shape of the cow and that you don't need it, it might as well be spherical. Now, a spherical cow falling on water would be a comical sight. Don't you think? Well, you might also consider that a lot of flailing legs and a panicked look on the face of a cow is funny.
(There's also a link there to Tom's old post on the point-cow approximation. That's three spherical cows this week...)
In related news, I'd like to ask for a moratorium on people starting really good physics blogs that I don't have time to read. Thank you.
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David Berenstein's example of how spherical cows might be "put to use" is not actually an example of how spherical cows might be put to use. In his example, there is no reason to assume the cow to be spherical. It might as well be cylindrical, conical, or even cow-shaped. The N=N+1 approximation does serve a purpose, it makes terms start dropping out of what can be large cumbersome equations and helps to provide insight into some systems.
I think we need some more work in finding problems where the spherical cow approximation actually has some impact on the problem. I suppose putting friction into our falling cow problem would be one way to do it, but I don't know that we'd be using a friction assumption that would be specifically taking into account the shape of the cow. Also, it seems like we'd have to make an approximation on the furriness of the cow, too.
@ Clark:
Soooo, you're suggesting they use "Assume a spherical, bald cow?" ;)
Are spherical cows the source of milk duds?
Clark,
I don't think it matters for this problem, because we're assuming the gravitational field is uniform. However, if the cow were falling from outer space (!), we would implicitly assume that it is spherical about its center of mass. If we treated the cow as any other shape, we'd have to calculate a very ugly integral of tidal forces, and we'd get about the same results.
Here is where you can get the book. http://www.amazon.com/Consider-Spherical-Cow-Environmental-Problem/dp/0…
So Minkowski, Poincare, and Einstein walk into a bar. The bartender looks at them and says: "consider a hyperspherical cow..."
i hereby promise to keep my physics blog crappy, infrequently updated, and without any real science content.
Sorry for being interesting, Chad. But thanks for noticing.
And to Clark, if you do consider air resistance on a falling cow, considering a spherical cow makes more sense, which is the example that I made in my post:
However, in certain situations, estimating the cow as a sphere with a characteristic radius might not be as ridiculous as it seems. For example, if the car were flying through the air (or standing in a strong wind, if flying spherical cows are too much for you to accept), the air resistance on a sphere the size of a cow would be a pretty accurate approximation.
I think that considering a point-mass cow makes more sense than a spherical cow. Unless you're trying to guess its moment of inertia...
But, Daryl, compressing a spherical cow to a point-mass cow may cause more destruction than those little black-holes the large hadron collider was feared to make.
Now we shall consider hypothetical Biology in a hyperbolic spaces. Consider a cow which has the shape of an n-manifold, i.e. a complete Riemannian n-manifold of constant sectional curvature -1.
Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and/or ends which are the product of a Euclidean (n-1)-manifold and the closed half-ray. The manifold is of finite volume if and only if its thick part is compact.
The hyperbolic structure on a finite volume hyperbolic n-manifold is unique by Mostow rigidity and so geometric invariants are in fact topological invariants.
Any questions? You, in the front row, when does the cow have finite volume? Well then, please listen when I'm talking...
Now, I'm handing out the homework assignment for you to work on this weekend. Problem #1: is it possible to have a 4-dimensional atom -- I mean with 4 spacial dimensions and 1 time dimension? Be sure to show your work. Class dismissed. Have a nice weekend.